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Function or a functional unit is defined as the association between a set of inputs and set of outputs. It can be defined as the relation of objects. When a function is defined there exists at least one value in the output set for value in an input set and if a function is one- one function then there exists exactly one value or one output for every input.

A function is of many types that are real functions, scalar functions, vector function, complex function, and a linear function. Linear function definition can be given as if it satisfies two main properties. First one among it is when f(a * x) is equals to a * f(x) where 'a' is a constant and 'x' is a variable and second property is given by f (x + y) is equals to f(x) + f(y). if a function satisfies these properties then it is said to be satisfying a linear function.

It is a single degree function and thus can also be considered as a polynomial function as it is represented by f (x) it is termed as a function of 'x'. A linear function is also a algebraic function in which a term is either in a form of product of constant or a constant or a single variable. For representation of a linear function if the coordinates are given then it can be expressed in the form of function and similarly opposite also exists that is if a function is given then coordinates can be determined. For example, if the coordinate is given as (4, 16) then function for this coordinate will be f (4) which is equals to 16. And graphically also it can be easily represented.

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Suppose we have the statement as follows:

If 5 is added to 3 times the number and sum is 26, then find the number.

Here we will proceed as follows:

Let the number be 'x'. So we say that three times of the number will be 3 * x. Thus the equation will be:

3 * x + 5 = 26,

Now to get the value of 'x', i.e. the number, we will keep the value of 'x' to one side of the equation and all constant values are taken to another side of the equation. So we get:

3 * x = 26 – 5,

Or we write:

3 * x = 21,

Now we will divide the value of right side of the equation i. e . 21 by the number 3 and the equation will be written as follows:

x = 21 / 3,

Or x = 7,

Now we will observe that required number will be 7. So if we will add 5 to three times of the number 7, we get:

7 * 3 + 5 = 21 + 5 = 26.

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A Function can be defined or explained as any event, action or task performed by any person for whom it is suited. Considering it under mathematical terms, a function is defined as the relation or association of any variable with the other in such a way that there exists a value or an output for each value of one variable defined.

For every input a particular output is defined. It can also be defined in terms of a set, in which for each of its element there exists a unique value. There are number of functions defined as complex functions, vector functions, real functions, scalar functions,linear functions and many other types of functions.

Now let us know about linear function. A linear function is defined as a function which satisfies the following given properties which are defined as is f (a*x) is equals to a* f(x) and second property is defined as f (x + y) is equals to f (x) + f (y) where ‘x’ and ‘y’ are used as variables for the functions and ‘a’ is a constant term that is multiplied with the function.

Function can be easily determined when the coordinates for any line are known. For example: If a coordinate is given as (4, 6) where 4 is said to be x coordinate and similarly 6 is y coordinate. Then, the function for this coordinate will be written as f(4) is equals to 6. Similarly if we know the function then the coordinates can be easily determined. Suppose, f(2) is given to be 10. Then the coordinate for this function will be determined as (2, 10) as the coordinates of linear function.

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It is also said to be linear because the graphs of these functions in the cartesian co- ordinate is straight line. We will understand it with the help of small example:

Suppose we have given some functions below which has graph that is a straight line.

F (p) = 2p + 4,

F (p) = p / 2 – 3.

It can also be written in following manner as:

f (p) = mp + b,

(y – y1) = m (x – x1),

0 = Ax + By + C.

In the case of vector Algebra a linear function means that a linear map that is a map between two vectors spaces where refers vector addition and scalar multiplication.

The linear functions are those functions ‘f’ which can be represented as,

f (p) = Kp, here ‘K’ is a matrix.

A function F (p) = mp + b is said to be a linear map if and only if value of b = 0.

The form y = mx + b is named as the slope intercept form of linear function. For a common graph (p, q) usually expressed as q = mp + b and in a formal function definition a linear function written as f (x) = mx + b. In this we can easily solve any function.

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